Task
An Olympic $400$ meter track is made up of two straight sides, each
measuring $84.39$ meters in length, and two semicircular curves with
a radius of $36.5$ meters as pictured below:
The picture is drawn to scale with one centimeter in the picture representing
$20$ meters on an actual olympic track. The one of the eight lanes which is closest to the center of the track is called the first lane.

What is the perimeter of the track, measured on the innermost part of the first lane?

Each lane on the track is $1.22$ meters wide. What is the perimeter of
the track measured
on the outermost part of the first lane?

In order to run the intended $400$ meters in a lap, how far away from the inside of the first lane would a runner need to be?
Below is an enlarged picture of one of the straight sections of the track with
the blue line representing the line around the track with perimeter exactly $400$ meters:
IM Commentary
In this problem geometry is applied to a $400$ meter track. The specifications
for building a track for the Olympics are very precise and are laid out on pages
$14$ and $15$ of
This task uses geometry to find the perimeter of the track. Students
may be surprised when their calculation does not give $400$ meters but
rather a smaller number. The teacher may wish to stop students at this
point for a discussion of how this can be. Hopefully students will have the idea
that if a runner's body were centered on the border of the track, the runner
would only be half on the track and hence considered out of bounds. The
next two parts of the problem give an idea where the runner needs to be
in order to be running $400$ meters in one lap. If the students have a track
at school they may want to measure out $30$ centimeters and see if it
is realistic to run around the track with their body centered on this line.
This task is further developed in ''Running around a track II'' where the
staggered starts for different lanes are considered. This task is ideal
for instruction but could be use for assessment as well. If used for instruction
purposes, the teacher may wish to ask students if the lane lines of the track are similar geometric shapes. This is a subtle question as the curved sections and the straight sections are similar but the entire shapes are not because there is a scale factor in the curves (with the outer lanes being radii of larger circles) while the straight sections are the same length for all lanes.
A simplified version of this task, without part (c) or with a different solution method, could be used in Grade 7 for 7.G.4, "Know the formulas for the area and circumference of a circle and use them to solve problems ...." And a more challenging modeling task for high school could remove some of the scaffolding and expect students to research the dimensions themselves.